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June 13, 2026

Exact Approximation in the Field of Formal Power Series

Key Points

The aim is to determine the Hausdorff dimension of the set of exactly ψ-approximable vectors over local fields with positive characteristic.
Adaptation of Bandi and de Saxce's method to the ultrametric setting
Analysis within the context of local fields of positive characteristic
Examination of rational functions in higher dimensions.
The Hausdorff dimension is computed for exactly ψ-approximable vectors over local fields.
This result extends the theorem of Zhang to higher dimensions.
It provides a function field analogue of a recent theorem in the real setting.

Abstract

In this article, we determine the Hausdorff dimension of the set of exactly ψ-approximable vectors over local fields of positive characteristic. This result is the function field analogue of a recent theorem of Bandi and de Saxce in the real setting BdS25, and extends the main theorem of Zhang Zha12 to higher dimensions. Our approach adapts the method of Bandi and de Saxce to the ultrametric setting, enabling us to overcome difficulties arising from the failure of the well-separatedness property for rational functions in higher dimensions.

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